(*  Title:      HOL/Hoare/SepLogHeap.thy
    Author:     Tobias Nipkow
    Copyright   2002 TUM
*)

section ‹Heap abstractions for Separation Logic›

text ‹(at the moment only Path and List)›

theory SepLogHeap
  imports Main
begin

type_synonym heap = "(nat → nat option)"

text‹‹Some› means allocated, ‹None› means
 . Address ‹0› serves as the null reference.›


subsection "Paths in the heap"

primrec Path :: "heap → nat → nat list → nat → bool"
where
  "Path h x [] y = (x = y)"
| "Path h x (a#as) y = (x≠0 ∧ a=x ∧ (∃b. h x = Some b ∧ Path h b as y))"

lemma [iff]: "Path h 0 xs y = (xs = [] ∧ y = 0)"
by (cases xs) simp_all

lemma [simp]: "x≠0 ==> Path h x as z =
 (as = [] ∧ z = x ∨ (∃y bs. as = x#bs ∧ h x = Some y & Path h y bs z))"
by (cases as) auto

lemma [simp]: "∧x. Path f x (as@bs) z = (∃y. Path f x as y ∧ Path f y bs z)"
by (induct as) auto

lemma Path_upd[simp]:
 "∧x. u ∉ set as ==> Path (f(u := v)) x as y = Path f x as y"
by (induct as) simp_all


subsection "Lists on the heap"

definition List :: "heap → nat → nat list → bool"
  where "List h x as = Path h x as 0"

lemma [simp]: "List h x [] = (x = 0)"
by (simp add: List_def)

lemma [simp]:
 "List h x (a#as) = (x≠0 ∧ a=x ∧ (∃y. h x = Some y ∧ List h y as))"
by (simp add: List_def)

lemma [simp]: "List h 0 as = (as = [])"
by (cases as) simp_all

lemma List_non_null: "a≠0 ==>
 List h a as = (∃b bs. as = a#bs ∧ h a = Some b ∧ List h b bs)"
by (cases as) simp_all

theorem notin_List_update[simp]:
 "∧x. a ∉ set as ==> List (h(a := y)) x as = List h x as"
by (induct as) simp_all

lemma List_unique: "∧x bs. List h x as ==> List h x bs ==> as = bs"
by (induct as) (auto simp add:List_non_null)

lemma List_unique1: "List h p as ==> ∃!as. List h p as"
by (blast intro: List_unique)

lemma List_app: "∧x. List h x (as@bs) = (∃y. Path h x as y ∧ List h y bs)"
by (induct as) auto

lemma List_hd_not_in_tl[simp]: "List h b as ==> h a = Some b ==> a ∉ set as"
apply (clarsimp simp add:in_set_conv_decomp)
apply(frule List_app[THEN iffD1])
apply(fastforce dest: List_unique)
done

lemma List_distinct[simp]: "∧x. List h x as ==> distinct as"
by (induct as) (auto dest:List_hd_not_in_tl)

lemma list_in_heap: "∧p. List h p ps ==> set ps ⊆ dom h"
by (induct ps) auto

lemma list_ortho_sum1[simp]:
 "∧p. [ List h1 p ps; dom h1 ∩ dom h2 = {}] ==> List (h1++h2) p ps"
by (induct ps) (auto simp add:map_add_def split:option.split)


lemma list_ortho_sum2[simp]:
 "∧p. [ List h2 p ps; dom h1 ∩ dom h2 = {}] ==> List (h1++h2) p ps"
by (induct ps) (auto simp add:map_add_def split:option.split)

end